A one-dimensional Keller–Segel equation with a drift issued from the boundary

Calvez, Vincent; Meunier, Nicolas; Voituriez, Raphael

Comptes Rendus. Mathématique, Tome 348 (2010) no. 11-12, pp. 629-634.

Nous étudions dans cette Note la dynamique d'un modèle unidimensionnel de type Keller–Segel posé sur une demi-droite. Dans le cas présent, la production du signal chimique est localisée sur le bord, au lieu d'être répartie à l'intérieur du domaine comme dans le cas classique. On démontre, sous des hypothèses convenables, la dichotomie suivante qui rappelle le système de Keller–Segel en dimension deux d'espace. Les solutions sont globales si la masse est sous-critique, elles explosent en temps fini si la masse dépasse la masse critique. Enfin, les solutions convergent vers un état d'équilibre lorsque la masse est égale à la valeur critique. Des méthodes d'entropie sont développées, dans le but d'obtenir des résultats de convergence quantitatifs. Cette Note est enrichie d'une brève introduction à un modèle plus réaliste (à nouveau unidimensionnel).

We investigate in this Note the dynamics of a one-dimensional Keller–Segel type model on the half-line. On the contrary to the classical configuration, the chemical production term is located on the boundary. We prove, under suitable assumptions, the following dichotomy which is reminiscent of the two-dimensional Keller–Segel system. Solutions are global if the mass is below the critical mass, they blow-up in finite time above the critical mass, and they converge to some equilibrium at the critical mass. Entropy techniques are presented which aim at providing quantitative convergence results for the subcritical case. This Note is completed with a brief introduction to a more realistic model (still one-dimensional).

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